diff --git a/src/2026-01-16/presentation.tex b/src/2026-01-16/presentation.tex index e191ce8..4a390aa 100644 --- a/src/2026-01-16/presentation.tex +++ b/src/2026-01-16/presentation.tex @@ -202,6 +202,67 @@ % TODO: +\begin{frame} + \frametitle{Zusammenfassung} + + \begin{columns}[t] + \column{\kittwocolumns} + \begin{greenblock}{Korrelationskoeffizient} + \vspace*{-6mm} + \begin{gather*} + \rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}} + \end{gather*} + \end{greenblock} + \begin{greenblock}{Kovarianz} + \vspace*{-6mm} + \begin{gather*} + \text{cov}(X,Y) = E(X\cdot Y) - E(X)E(Y) + \end{gather*} + \end{greenblock} + \begin{greenblock}{Randdichte} + \vspace*{-6mm} + \begin{gather*} + f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) dy + \end{gather*} + \end{greenblock} + \column{\kitfourcolumns} + \begin{greenblock}{Transformationssatz} + \vspace*{-6mm} + \begin{gather*} + X = h_1(U,V), \hspace{5mm} Y = h_2(U,V) \\[2mm] + \mathcal{J} = + \begin{pmatrix} + \frac{\displaystyle \partial}{\displaystyle + \partial u}x & \frac{\displaystyle + \partial}{\displaystyle \partial v}x \\[2mm] + \frac{\displaystyle \partial}{\displaystyle + \partial u}y & \frac{\displaystyle + \partial}{\displaystyle \partial v}y + \end{pmatrix} + = \begin{pmatrix} + \frac{\displaystyle \partial}{\displaystyle + \partial u}h_1(u,v) & \frac{\displaystyle + \partial}{\displaystyle \partial v}h_1(u,v) \\[2mm] + \frac{\displaystyle \partial}{\displaystyle + \partial u}h_2(u,v) & \frac{\displaystyle + \partial}{\displaystyle \partial v}h_2(u,v) + \end{pmatrix} \\[3mm] + f_{U,V}(u,v) = \lvert + \text{det}(\mathcal{J}) \rvert + \cdot f_{X,Y} \big(h_1(u,v),h_2(u,v)\big) + \end{gather*} + \end{greenblock} + \begin{greenblock}{Erwartungswert \& Varianz} + \vspace*{-6mm} + \begin{align*} + V(X) &= E\big( (X - E(X))^2 \big) = E(X^2) - E^2(X) \\ + E(X) &= \int_{-\infty}^{\infty} x f_X(x) dx \\ + E(g(X)) &= \int_{-\infty}^{\infty} g(x) f_X(x) dx + \end{align*} + \end{greenblock} + \end{columns} +\end{frame} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Aufgabe}